Chomp

👥 2 players 📍 Indoor📍 Anywhere ⚡ Calm 🧩 Moderate ⏱ 10-30 minutes 🎂 Ages 6+

Quick Pitch

Chomp is a two-player game where you take turns eating squares from a chocolate bar grid — but whoever is forced to eat the poisoned corner square loses.

Hook

Imagine a chocolate bar drawn on paper as a grid of squares. On your turn, pick any square — and that square plus everything above it and to its right gets eaten away. The one square nobody wants is the poisoned square in the lower-left corner. Mathematicians have proven that the first player always has a winning move, but for boards bigger than about 2×3, nobody has ever figured out what that winning move actually is. You're playing a game with a known solution that no one can find.

Equipment Needed

  • Sheet of paper
  • Pencil or pen
  • Ruler (helpful for drawing grid)

Setup

  1. Create Grid (Rectangular Chocolate Bar):

    • Standard: 3×4 grid (12 squares)
    • Simple: 2×2 grid (4 squares)
    • Complex: 4×5 or larger
    • Can be any rectangular dimensions

  2. Mark Poisoned Square: Mark lower-left corner (or agreed corner) as poisoned

Example 3×4 grid:

. . . .
. . . .
P . . .

(P = Poisoned square in lower-left corner)

  1. Decide Turn Order: Player 1 goes first

Rules

Objective

Force opponent to take the poisoned square. The player forced to eat the poisoned square loses.

Gameplay

  1. Players alternate turns
  2. On each turn, a player selects any square that hasn't been eaten
  3. That square and ALL squares above it, to its right, and diagonally up-right are "eaten" (removed)
  4. The remaining grid must form a rectangular shape

Eating Example

Starting grid with poisoned square (P):

A B C D
E F G H
P I J K

If you eat square J:

  • J and all squares above-right are eaten
  • Above J: G, K, H
  • Right of J: K (but already counted)

Result:

A B . .
E F . .
P I . .

Remaining rectangle: 2×2 with positions A, B, E, F, P, I (top-left becomes 2 rows × 2 columns)

Winning Condition

The player forced to eat the poisoned square (P) loses immediately. A player unable to avoid eating P has lost.

Expert Player

Tips

Important Facts About Chomp:

  1. First Player Always Has Winning Move:

    • Proven mathematically (strategy stealing argument)
    • But strategy is unknown for most board sizes

  2. Square Eating:

    • When you eat a square, all squares in the rectangle formed by (current square) and (upper-right corner) are eaten
    • Must maintain rectangular grid

  3. Strategy Stealing Proof:

    • If second player had winning strategy, first player could "steal" it
    • But first player can always move, so second player can't have winning strategy
    • Therefore, first player has winning strategy

  4. But No One Knows What It Is:

    • For even small boards (4×5), no known winning strategy
    • Computers have analyzed small boards
    • It's an interesting mathematical mystery

Practical Play Tips:

  1. Keep Grid Balanced:

    • Try to maintain symmetric or nearly symmetric positions
    • Asymmetric positions disadvantage the player to move

  2. Avoid Large Bites:

    • Eating from middle of grid is safer than corners
    • Large moves can create winning positions for opponent

  3. Control Corners:

    • Corner positions are dangerous
    • Eating corners gives opponent fewer options

  4. Visualize Results:

    • Before eating, visualize resulting rectangle
    • Think 2-3 moves ahead
    • Plan towards opponent's mistakes

  5. Key Positions (for small boards):

    2×2 Board (with poison at bottom-left):

    A B
P C
    • First player eats B or C
    • If eat B → leaves P alone, second player must take it (first wins)
    • If eat C → leaves A, B, P in a row, second player eats A or B, first eats remaining, second takes P (first wins)

    2×3 Board (with poison at bottom-left):

    • First player wins by eating top-right
    • Creates positions where any second-player move loses

  6. Symmetry Strategy:

    • Try to mirror opponent's moves (eating symmetric square)
    • Works only if opponent didn't start with perfect move
    • If position is already near-symmetric, you might be losing

Variations

  • Larger Boards: 5×5, 6×6, or other sizes (harder to analyze)
  • Different Poisoned Square: Poison at different corner or position (changes strategy)
  • 3D Chomp: 3×3×3 cube (much harder to visualize and analyze)
  • Infinite Chomp: Theoretical game on infinite grid
  • Reverse Chomp: Taking the poisoned square WINS instead of losing
Learn More — History & Origins

History & Origins

Chomp was invented in 1974 by American mathematician David Gale, who presented it as an example of a "strategy-stealing argument" — a technique for proving that a first-player winning strategy exists without actually finding it. The argument goes like this: suppose the second player had a winning strategy. Whatever their first response to any opening move, the first player could have simply played that exact move themselves at the start. So if the second player had a winning strategy, the first player could steal it. Contradiction. Therefore, the first player must have a winning strategy.

This proof tells you the winning strategy exists for any rectangular board (with the poisoned square in the corner). What it can't tell you is what that strategy looks like. For boards larger than about 2×4, no one has managed to enumerate it — the number of possible game positions grows too fast. Chomp thus became a famous example in combinatorial game theory of a game that is provably determined but practically unsolved.

Cultural Context

Chomp regularly appears in introductory combinatorics and game theory courses because the strategy-stealing argument is both simple enough to explain to newcomers and surprising enough to stick. The idea that you can prove something exists without constructing it is one of the stranger corners of mathematics, and Chomp makes that strangeness concrete and playable.

As a game, it has a satisfying arc: early moves feel free and open, mid-game positions get increasingly constrained, and the endgame often comes down to a chain of forced moves where one player is herded toward the poison square. Players who've never heard of the strategy-stealing argument often rediscover it independently — noticing that the first player seems to keep winning and wondering why.

See Also

Small Board Solutions

Computers have analyzed small boards:

  • 2×2: First player wins
  • 2×3: First player wins
  • 2×4: First player wins
  • 3×3: First player wins
  • 3×4: First player wins (but strategy unknown)
  • 4×4: First player wins (but strategy unknown)

For larger boards, strategy remains unknown despite computer analysis.

Mathematical Significance

Chomp represents an interesting frontier in combinatorial game theory:

  1. Proven but Unknown: First player has winning strategy, but no one can articulate it (beyond brute-force analysis)

  2. Computational Complexity: Finding optimal strategy requires examining many positions

  3. Strategy Stealing Argument: Shows first player advantage without revealing strategy

  4. Open Problem: Finding winning strategy for general rectangular boards remains unsolved